###### Covariance and Correlation

A frequency distribution refers to the presentation of statistical data in a tabular format to simplify data analysis. In a frequency distribution, data is subdivided into groups or intervals.

The standard procedure for constructing a frequency distribution involves the following steps:

- Determine the number of classes one wishes to have. 5 – 20 is always a good number.
- Determine the interval size. To do this, the range and the number of classes should guide an analyst. The range is the difference between the smallest and the largest observations. In the case of a fractional result, an analyst should take the next higher whole number as the size of the interval.
- Determine the starting point. It could be the lower limit of the lowest observation or a convenient value just below the lower limit.
- Add the class interval to the starting point to get the second lower limit. This process should be repeated.
- List the lower limits in a vertical column alongside the upper-class limits.
- Complete the table by counting the number of observations that fall under each class.

Points to note:

- Classes should be mutually exclusive and have similar widths.
- The last class includes the maximum value.
- All class intervals should be tabulated even if they have zero observations.
- The sum of the frequencies should be equal to the number of observations.
- Tally bars offer a convenient tool for the visual presentation of the number of observations in each.

You have been given the following data showing the percentage returns that certain classes of investment offer in a year. Use the data to construct a frequency distribution table.

$$ \begin{array}{c|c|c|c|c} \text{-10%} & \text{2%} & \text{32%} & \text{-28%} & \text{25%} \\ \hline \text{-25.60%} & \text{4%} & \text{11%} & \text{-14%} & \text{15%} \\ \hline \text{23%} & \text{13%} & \text{6%} & \text{-2.70%} & \text{8%} \\ \hline \text{12%} & \text{28%} & \text{17.50%} & \text{5.80%} & \text{20%} \\ \hline \text{4.60%} & \text{17%} & \text{-3.90%} & \text{22.40%} & \text{15%} \\ \end{array} $$

**Solution**

- We have 25 observations in total. First of all, we will sort the data in ascending order.
- After this, we will calculate the range of the data, where Range = Maximum value − Minimum value = 32% – (-28%) = 60%.
- We will then determine an interval width of 10%.
- The lowest return intervals will be -30% ≤ Rt < -20%, while the highest one will be 30% ≤ Rt < 40%.
- Then, we will count the observations in each interval.

$$ \begin{array}{c|c|c} \textbf{Interval} & \textbf{Tally} & \textbf{Frequency} \\ \hline -30\% \leq R_t < -20\% & \text{II} & \text{2} \\ -20\% \leq R_t < -10\% & \text{I} & \text{1} \\ -10\% \leq R_t < 0\% & \text{III} & \text{3} \\ 0\% \leq R_t <10\% & \text{IIIIII} & \text{6} \\ 10\% \leq R_t < 20\% & \text{IIIIIII} & \text{7} \\ 20\% \leq R_t < 30\% & \text{IIIII} & \text{5} \\ 30\% \leq R_t < 40\% & \text{I} & \text{1} \\ \textbf{Total} & \text{} & \textbf{25} \\ \end{array} $$

**Absolute frequency** is the actual number of observations in a given interval.

**Relative frequency** refers to the percentage of observations falling within a given class. It is calculated by dividing the absolute frequency of each return interval by the total number of observations. Using our earlier example when we introduced the frequency distribution table, we could come up with the relative frequency for each interval using the formula below:

$$\text{Relative Frequency}=\frac{\text{Absolute frequency}}{\text{Total frequency}}$$

Where \(\text{Total frequency}\) is the total number of observations.

$$

\begin{array}{c|c|c|c}

\textbf { Interval } & \textbf { Tally } & \textbf { Frequency } & \textbf { Relative Frequency } \\

\hline-30 \% \leq \mathrm{R}_{\mathrm{t}} \leq-20 \% & \text { II } & 2 & \frac{2}{25}=8 \% \\

-20 \% \leq \mathrm{R}_{\mathrm{t}} \leq-10 \% & \text { I } & 1 & \frac{1}{25}=4 \% \\

-10 \% \leq \mathrm{R}_{t} \leq 0 \% & \text { III } & 3 & \frac{3}{25}=12 \% \\

0 \% \leq \mathrm{R}_{t} \leq 10 \% & \text { IIIII } & 6 & \frac{6}{25}=24 \% \\

10 \% \leq \mathrm{R}_{t} \leq 20 \% & \text { IIIIII } & 7 & \frac{7}{25}=28 \% \\

20 \% \leq \mathrm{R}_{t} \leq 30 \% & \text { IIII } & 5 & \frac{5}{25}=20 \% \\

30 \% \leq \mathrm{R}_{t} \leq 40 \% & \text { I } & 1 &\frac{1}{25}=4 \% \\

\text { Total } & & 25 & \frac{25}{25}=100 \%

\end{array}

$$

In the above table, the absolute frequency of the 1^{st} interval is 2. Similarly, the relative frequency of the 1^{st} interval is 8%. The same applies to other intervals.

**Cumulative absolute frequency** is the sum of the absolute frequencies, including the given interval.

**Cumulative relative frequency** similarly sums up the relative frequencies up to and including the given relative frequency.

$$

\begin{array}{c|c|c|c|c|c}

\textbf { Interval } & \textbf { Tally } & \textbf { Frequency } & \textbf { Relative Frequency } & \begin{array}{c}

\textbf { Cumulative Absolute } \\

\textbf { Frequency }

\end{array} & \begin{array}{c}

\textbf { Cumulative Relative } \\

\textbf { Frequency }

\end{array} \\

\hline-30 \% \leq R \leq-20 \% & \text { II } & 2 & \frac{2 }{25}=8 \% & 2 & 8 \% \\

-20 \% \leq R_{4} \leq-10 \% & \text { I } & 1 & \frac{1 }{25}=4 \% & 3 & 12 \% \\

-10 \% \leq R_{1} \leq 0 \% & \text { III } & 3 & \frac{3}{ 25}=12 \% & 6 & 24 \% \\

0 \% \leq R_{1} \leq 10 \% & \text { IIIII } & 6 & \frac{6}{25}=24 \% & 12 & 48 \% \\

10 \% \leq R \leq 20 \% & \text { IIIII } & 7 & \frac{7}{25}=28 \% & 19 & 76 \% \\

20 \% \leq R \leq 30 \% & \text { IIII } & 5 & \frac{5}{25}=8 \% & 24 & 96 \% \\

30 \% \leq R \leq 40 \% & \text { I } & 1 & \frac{1}{25}=4 \% & 25 & 100 \% \\

\text { Total } & & 25 & \frac{25}{25}=100 \% & &

\end{array}

$$

In the above table, cumulative absolute frequency is the sum of the absolute frequencies up to and including the given interval. The cumulative relative frequency similarly sums up the relative frequencies up to and including the given relative frequency.

Question 1The class frequency divided by the total number of observations is

most likelycalled:

- Relative frequency.
- Percentage frequency.
- Cumulative relative frequency.

SolutionThe correct answer is

A.Relative frequency refers to the percentage of observations falling within a given class. It is calculated by dividing the absolute frequency of each return interval by the total number of observations.

C is incorrect. Cumulative relative frequency sums up the relative frequencies up to and including the given relative frequency.

B is incorrect.There is no such term as percentage frequency.

Question 2The number of tally sheet count for each value or a group is

most likelyknown as:

- Class limit.
- Frequency.
- Class width.

SolutionThe correct answer is

B.The frequency of a class is the number of data entries in the class.

A is incorrect. Each class will have a “lower-class limit” and an “upper-class limit”, which are the lowest and highest numbers in each class.

C is incorrect. The “class width” is the distance between the lower limits of consecutive classes.